Consider the following classical construction:

Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space of alternating two-forms. Then the equation $a\wedge a\wedge a=0$, $a\in V^4$ is homogeneous of degree three and hence defines a cubic surface in ${\mathbb P}V^4$.

Question. Can every real cubic surface be obtained by the above construction? If yes, is the set of corresponding representations for each cubic connected? I would be grateful for a reference if there is one.

I am primarily interested in real case but if you only can comment on the complex case, this would be interesting for me as well.